Simple Cells Thurs. Jan. 25, 2018 1 Recall last lecture: DOG - PowerPoint PPT Presentation

COMP 546 Lecture 5 Orientation Selection 1: Simple Cells Thurs. Jan. 25, 2018 1

Recall last lecture: DOG (lateral inhibition), “cross correlation” image - - + - - - DOG 2

Example: an edge image 𝐽 𝑦, 𝑧 𝐽 𝑦 𝑦 3

Example: an edge image - - + - - - - - - - + + - - - - - - 𝐽 𝑦, 𝑧 - - + - - - DOG - - + - - - - - + - - - 𝐸𝑃𝐻 ⨂ 𝐽 𝑦, 𝑧 𝑦 4

Mach Bands Are they the result of lateral inhibition in the retina ? 5

ASIDE: Mach bands are well known problem for interpreting x-ray images. Very subtle changes in dark-bright must be detected and the visual system is often fooled. 6

Retinal ganglion cells encode image differences : • spectral (wavelength l ) , “chromatic” • spatial (x,y) • temporal (t) -- will cover this next week • spectral-spatial ( l, x, y) - Assignment 1 • spectral-spatio-temporal ( l, x, y, t) - omit 7

Assignment 1 R+ G- R+ G- R- G+ Single opponent cells Double opponent cells 8

Early visual pathway: retina to cortex (V1) Lateral geniculate nucleus (LGN) 9

Left visual field Right visual field (LGN) 10

Polar coordinates on the retina Vertical meridian Horizontal meridian 11

Retinotopic Map retina Definition: Cells in a visual area are spatially arranged in a retinotopic map if physically adjacent cells in that area have adjacent receptive fields (and hence encode image in adjacent regions of the retina) Some visual area in the brain e.g. LGN, V1 12

retina Layers 3, 4, 5, 6 LGN (small receptive fields, color opponency) Layers 1 and 2 (bigger receptive fields time dependent, no color opponency) V1 13

Polar coordinates in primary visual cortex (V1) - right visual field - 14

functional magnetic resonance imaging (fMRI) 15

What do cells in V1 encode ? (Hubel & Wiesel 1959) The moving slide (see 35 sec and on...) http://www.youtube.com/watch?v=IOHayh06LJ4 3 minutes of exploration: https://www.youtube.com/watch?v=Cw5PKV9Rj3o 16

“Simple Cell” Temporal effects to be discussed in lecture 7 17

V1 Orientation Tuning Curve 18

retina LGN V1 - - - - - - + - - - + - - - - + + - - - - - - - - - + - - - - + - - - - - - R+ G- Hubel and Wiesel R- G+ suggested this mechanism for elongated receptive field profile of V1 simple cell 19

Model of orientation selectivity in V1 - + - - + “Line Detector” “ Edge Detector ” - + - - + (even) (odd) - + - - + - + - - + 𝑀 = 𝑔 𝑦 − 𝑦 0 , 𝑧 − 𝑧 𝐽 𝑦, 𝑧 0 𝑦,𝑧 Cell centered at 𝑦 0 , 𝑧 0 20

Cell response model: half-wave rectification Response (spike rate) 𝑀 Quasi-linear : cell response is linear over some range. 21

𝑀 How to encode the negative values of ? (similar idea to last lecture) + - + - + - Line Detector + - + - + - (even) + - + - + - + - + - + - - + + - - + + - Edge Detector - + + - (odd) - + + - 22

“Gabor” function: classical model of simple cell Line (even) Edge (odd) 23

1D Cosine Gabor 𝑑𝑝𝑡𝑗𝑜𝑓 𝐻𝑏𝑐𝑝𝑠 𝑑𝑝𝑡𝑗𝑜𝑓 ∗ = 𝐻𝑏𝑣𝑡𝑡𝑗𝑏𝑜 24

1D Sine Gabor 𝑡𝑗𝑜𝑓 𝐻𝑏𝑐𝑝𝑠 𝑡𝑗𝑜𝑓 ∗ = 𝐻𝑏𝑣𝑡𝑡𝑗𝑏𝑜 25

(Sampled) Cosine cos( 2𝜌 𝑂 𝑙 𝑦 𝑦) 𝑙 𝑦 is spatial frequency 𝑓. 𝑕. 𝑙 𝑦 = 8 𝑂 = 256 26

1D Cosine Gabor cos(2𝜌 𝑂 𝑙 𝑦 𝑦) ∗ = 𝑓 − 𝑦 2 1 2𝜏 2 2𝜌 𝜏 27

1D Sine Gabor sin(2𝜌 𝑂 𝑙 𝑦 𝑦) ∗ = 𝑓 − 𝑦 2 1 2𝜏 2 2𝜌 𝜏 28

2D cosine cos 2𝜌 𝑂 (𝑙 𝑦 𝑦 + 𝑙 𝑧 𝑧) 𝑓. 𝑕. 𝑙 𝑦 = 4 𝑙 𝑧 = 0 𝑂 = 256 29

2D sine sin 2𝜌 𝑂 (𝑙 𝑦 𝑦 + 𝑙 𝑧 𝑧) 𝑓. 𝑕. 𝑙 𝑦 = 8 𝑙 𝑧 = 2 𝑂 = 256 30

model of simple cell: 2D Gabor cos 2𝜌 𝐻 𝑦, 𝑧, 𝜏 𝑂 (𝑙 𝑦 𝑦 + 𝑙 𝑧 𝑧) 𝑓. 𝑕. 𝑙 𝑧 = 0 sin 2𝜌 𝐻 𝑦, 𝑧, 𝜏 𝑂 (𝑙 𝑦 𝑦 + 𝑙 𝑧 𝑧) 𝑓. 𝑕. 𝑙 𝑧 = 0 31

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• What is the response of a family of Gabor cells to a single image ? e.g. Consider shifted versions of the Gabor cell. • What is the response of a single Gabor cell to a parameterized family of images ? e.g. thin white line at different positions in receptive field 33

What is the response of a family of Gabor cells to a single image? 34

cross correlation with (four) cosine Gabors 35

cross correlation with (four) sine Gabors 36

• What is the response of a family of Gabor cells to a single image ? e.g. Consider shifted versions of the Gabor cell. • What is the response of a single Gabor cell to a parameterized family of images ? e.g. thin white line at different positions in receptive field 37

𝑀 ≡ 𝑑𝑝𝑡𝐻𝑏𝑐𝑝𝑠 𝑦 , 𝑧 𝐽 𝑦, 𝑧; 𝑦 𝑥ℎ𝑗𝑢𝑓 𝑚𝑗𝑜𝑓 𝑦 ,𝑧 38

𝑀 ≡ 𝑑𝑝𝑡𝐻𝑏𝑐𝑝𝑠 𝑦 , 𝑧 𝐽 𝑦, 𝑧; 𝑦 𝑥ℎ𝑗𝑢𝑓 𝑚𝑗𝑜𝑓 𝑦 ,𝑧 cos 2𝜌 Non-zero only at 𝑦 position 𝐻 𝑦, 𝑧, 𝜏 𝑂 (𝑙 𝑦 𝑦) of vertical line 39

𝑀 ≡ 𝑡𝑗𝑜𝐻𝑏𝑐𝑝𝑠 𝑦 , 𝑧 𝐽 𝑦, 𝑧; 𝑦 𝑥ℎ𝑗𝑢𝑓 𝑚𝑗𝑜𝑓 𝑦 ,𝑧 𝑡𝑗𝑜 2𝜌 Non-zero only at 𝑦 position 𝐻 𝑦, 𝑧, 𝜏 𝑂 (𝑙 𝑦 𝑦) of vertical line 40

Gaussian envelope (discuss next lecture) 41